Electromagnetic Metamaterials are periodically arranged artificial structures that show peculiar behaviors such as negative refraction not seen in natural materials[1]}, [2]}. The photonic atoms, or the element structures of the metamaterials, are typically much smaller in size than the working wavelengths such that the metamaterials can be considered to be homogeneous and macroscopic parameters such as electrical permittivity and magnetic permeability can be used to describe the electromagnetic properties of the metamaterials [3]}. By carefully engineering the photonic atoms, both the permittivity and permeability can be made negative such that negative refraction can be achieved from the metamaterial [4]}, [5]}. The most widely used structures for this purpose is the composite of short metallic wires and split-ring resonators (SRRs) [6]}, [7]}. While the array of short wires gives a negative permittivity in the wide frequency range below the effective plasma frequency, the array of SRRs gives a negative permeability in the narrow frequency range just above the resonance frequency so that the effective index of refraction can be negative in a narrow frequency band. While most of the research in this area is in a linear regime, where the electromagnetic responses are independent of the external fields, some effort has been made to study the nonlinear effects of the metamaterials, especially the nonlinear tunability of the SRRs [8-12]. The SRRs are essentially LC resonators and the resonance frequency is determined by the geometry of the rings. To tune the magnetic responses of the SRRs, extra components or materials need to be introduced into the SRRs.
